Space and time renormalization in phase transition dynamics

When a system is driven across a quantum critical point at a constant rate, its evolution must become nonadiabatic as the relaxation time $\tau$ diverges at the critical point. According to the Kibble-Zurek mechanism (KZM), the emerging post-transition excited state is characterized by a finite correlation length $\widehat{\xi }$ set at the time $\widehat{t}=\widehat{\tau }$ when the critical slowing down makes it impossible for the system to relax to the equilibrium defined by changing parameters. This observation naturally suggests a dynamical scaling similar to renormalization familiar from the equilibrium critical phenomena. We provide evidence for such KZM-inspired spatiotemporal scaling by investigating an exact solution of the transverse field quantum Ising chain in the thermodynamic limit.

Figure 1

Schematic illustration of the quantum Kibble-Zurek mechanism. The Hamiltonian is driven by a linear quench (3) across the critical point at $t=0$. The energy gap (2) closes like ${\left|\epsilon \right|}^{z\nu }$, where $z$ and $\nu$ are universal exponents, and the transition rate (4) diverges at the critical point; hence the evolution cannot be adiabatic between $-\widehat{t}$ and $+\widehat{t}$. This figure shows the gap for $z=1$ and $\nu =1/2$, i.e., the mean-field scalings. In the quantum Ising model, $z=\nu =1$ and the gap closes linearly as $t\to 0$.

Figure 2

(a) The ferromagnetic correlation function in the quantum Ising chain (18), $C_R^{xx}=\langle {\sigma }_n^x{\sigma }_{n+R}^x\rangle -\langle {\sigma }_n^x\rangle \langle {\sigma }_{n+R}^x\rangle$, as a function of the rescaled distance $R/\widehat{\xi }$ at different rescaled times $t/\widehat{t}=-2,...,3$. Here, ${\sigma }_n^x$ is the $x$ component of a spin $1/2$ at the lattice site $n$ and $\langle {\sigma }_n^x\rangle$ is the ferromagnetic magnetization (order parameter). The quench time is ${\tau }_Q=1024$. (b) A correlation range $\langle R\rangle$ as a function of time. We define $\langle R\rangle$ as the distance $R$ where the correlator $C_R^{xx}$ falls below $0.1$. This cutoff is somewhat arbitrary. The behavior of $\langle R\rangle$ is similar for other cutoffs of this order. It is selected to provide the size of domains that choose the same broken symmetry state (rather than as an estimate of the correlation length, which is defined for much smaller values of $C_R^{xx}$). The range $\langle R\rangle$ increases several times between $-\widehat{t}$ and $\widehat{t}$ before it freezes after $\widehat{t}$. Similar plots to (a) and (b) were made in Ref. [66]. (c) The velocity $v=d\langle R/\widehat{\xi }\rangle /d(t/\widehat{t})$—the derivative of the plot in (b)—as a function of the rescaled time. Since in the Ising chain $z=1$ and $\widehat{\xi }=\widehat{t}$, it is also the velocity in the standard units, $v=d\langle R\rangle /dt$. Near $t/\widehat{t}\approx 0$, the correlation is spreading with nearly twice the speed of quasiparticles, $c=2$, at the critical point, suggesting that “sound cones” (in analogy with “light cones”) are responsible for the size of the domains.

Figure 3

The Landau-Zener modes (33) with different $k$ pass through their anticrossings at different $g_{\mathrm{AC}}\left(k\right)=cos\left(k\right)$. In the adiabatic limit, the size of the nonadiabatic regime $\widehat{k}\simeq 1/\sqrt{{\tau }_Q}$ becomes small, $1-g_{\mathrm{AC}}\left(\widehat{k}\right)\simeq 1/{\tau }_Q$ becomes much less than $1-g\left(\widehat{t}\right)\simeq 1/\sqrt{{\tau }_Q}$, and we can safely approximate $g_{\mathrm{AC}}\left(k\right)\approx 1$, as if all of the anticrossings took place simultaneously at the critical $g=1$. This is the essence of the approximation between Eqs. (40) and (41).

Figure 4

Quadratic fermionic correlators in Eqs. (42) and (43). The first row shows the real ${\alpha }_R=\langle c_R{c}_0^{†}\rangle$. The second and third rows show the real and imaginary parts of ${\beta }_R=\langle c_R{c}_0\rangle$, respectively. The left, middle, and right columns show the correlators before ($t/\widehat{t}=-1$), at ($t/\widehat{t}=0$), and after ($t/\widehat{t}=+1$) the critical point, respectively. Different colors of the plots correspond to different quench times ${\tau }_Q$. All plots are rescaled: they are presented as a function of the rescaled distance $R/\widehat{\xi }=R/\sqrt{{\tau }_Q}$ and the correlators are multiplied with $\widehat{\xi }=\sqrt{{\tau }_Q}$. At large distance, $R\gg 1$, the plots in all nine panels collapse asymptotically with increasing ${\tau }_Q$, demonstrating the scaling hypotheses (51) and (53) for large enough ${\tau }_Q$. The collapse does not happen at short distance, $R\approx 1$, where the microscopic lattice constant remains a relevant scale of length and the scaling hypothesis is not expected to hold. The collapsed plots for large ${\tau }_Q$ are the scaling functions $F_{\alpha }(t/\widehat{t},R/\widehat{\xi })$ and $F_{\beta }(t/\widehat{t},R/\widehat{\xi })$.

Figure 5

Top: the density of quasiparticle excitations in Eq. (54) as a function of the rescaled time $t/\widehat{t}$. Different colors of the plots correspond to different quench times ${\tau }_Q$. Each plot reaches up to $t/\widehat{t}=\sqrt{{\tau }_Q}$, corresponding to zero transverse field $g=0$. The inset is a focus on the nonadiabatic stage $t/\widehat{t}=-1...1$. Bottom: the density of excitation energy in Eq. (55) as a function of $t/\widehat{t}$. In both panels, the plots collapse asymptotically with increasing ${\tau }_Q$, demonstrating the scaling hypotheses (57) for large enough ${\tau }_Q$. The collapsed plots for large ${\tau }_Q$ are the scaling functions $F_n(t/\widehat{t})$ and $F_W(t/\widehat{t})$ in Eq. (57). The same quantities with a different rescaling of time can be seen in Fig. 6.

Figure 6

(a) The density of quasiparticle excitations as a function of $g$. Quasiparticles are excited at the two critical points, $g=\pm 1$. For any fixed $g$, except near $g=\pm 1$, their density scales as $n/N\sim {\widehat{\xi }}^{-1}$ (i.e., the typical distance between the kinks is set by $\widehat{\xi }$). (b) The density of excitation energy as a function of $g$. For any fixed $g$, it scales like $W/N\sim {\widehat{\xi }}^{-1}$, except near the gapless critical points $g=\pm 1$. Apart from the immediate vicinity of the critical points, the evolution is adiabatic. That is, the increase of energy is caused by the increase in the separations between the occupied energy levels. Thus, when the Ising chain is still in its ground state [leftmost segment in (b)], $W=0$, but after the excitation caused by the passage through the critical point at $g=1$, the slope increases and remains constant until $g=-1$. Additional excitations double the slope in the rightmost segments of (b). Indeed, these slopes are set by the quasiparticle densities seen in (a).

Figure 7

Spin-spin correlation functions in Eq. (59). The left, middle, and right columns show the correlators before ($t/\widehat{t}=-1$), at ($t/\widehat{t}=0$), and after ($t/\widehat{t}=+1$) the critical point, respectively. The first row shows the strongest ferromagnetic correlator $C_R^{xx}$, the second and third ones show $C_R^{yy}$ and $C_R^{xy}$, respectively, and the bottom one shows the transverse $C_R^{zz}$. Different colors of the plots correspond to different quench times ${\tau }_Q$. All plots are rescaled: they are presented as a function of the rescaled distance $R/\widehat{\xi }=R/\sqrt{{\tau }_Q}$ and the correlators are multiplied with ${\widehat{\xi }}^{{\mathrm{\Delta }}_a+{\mathrm{\Delta }}_b}={\left(\sqrt{{\tau }_Q}\right)}^{{\mathrm{\Delta }}_a+{\mathrm{\Delta }}_b}$. At large distance, $R\gg 1$, the plots in all 12 panels collapse asymptotically with increasing ${\tau }_Q$, demonstrating the space-time scaling (59) for large enough ${\tau }_Q$. The collapse does not happen at short distance, $R\approx 1$, where the microscopic lattice constant remains a relevant scale of length and the scaling hypothesis is not expected to hold. The collapsed plots for large ${\tau }_Q$ are the scaling functions $F_C^{ab}(t/\widehat{t},R/\widehat{\xi })$ in Eq. (60).

Figure 8

Mutual information in Eq. (63). The left, middle, and right panels show the mutual information before ($t/\widehat{t}=-1$), at ($t/\widehat{t}=0$), and after ($t/\widehat{t}=+1$) the critical point, respectively. Different colors of the plots correspond to different quench times ${\tau }_Q$. All plots are rescaled: they are presented as a function of the rescaled distance $R/\widehat{\xi }=R/\sqrt{{\tau }_Q}$ and the mutual information is multiplied with ${\widehat{\xi }}^{4{\mathrm{\Delta }}_a}={\left(\sqrt{{\tau }_Q}\right)}^{1/2}$. The plots in all three panels collapse asymptotically with increasing ${\tau }_Q$, demonstrating the space-time scaling (65) for large enough ${\tau }_Q$. The collapsed plots for large ${\tau }_Q$ are the scaling function $F_I(t/\widehat{t},R/\widehat{\xi })$ in Eq. (65).

Figure 9

Quantum discord in Eq. (66). The left, middle, and right panels show the discord before ($t/\widehat{t}=-1$), at ($t/\widehat{t}=0$), and after ($t/\widehat{t}=+1$) the critical point, respectively. Different colors of the plots correspond to different quench times ${\tau }_Q$. All plots are rescaled: they are presented as a function of the rescaled distance $R/\widehat{\xi }=R/\sqrt{{\tau }_Q}$ and the discord is multiplied with ${\widehat{\xi }}^{4{\mathrm{\Delta }}_a}={\left(\sqrt{{\tau }_Q}\right)}^{1/2}$. The plots in all three panels collapse asymptotically with increasing ${\tau }_Q$, demonstrating the space-time scaling (70) for large enough ${\tau }_Q$. The collapsed plots for large ${\tau }_Q$ are the scaling function $F_{\delta }(t/\widehat{t},R/\widehat{\xi })$ in Eq. (70).

Figure 10

Entanglement entropy in Eq. (74). The left, middle, and right panels show the entropy before ($t/\widehat{t}=-1$), at ($t/\widehat{t}=0$), and after ($t/\widehat{t}=+1$) the critical point, respectively. Different colors of the plots correspond to different quench times ${\tau }_Q$. All plots are rescaled: they are presented as a function of the rescaled size of the block, $L/\widehat{\xi }=L/\sqrt{{\tau }_Q}$, and the entropy is divided by $S_{\infty }(t/\widehat{t})=\frac{c}{3}lnk\widehat{\xi }=\frac{c}{3}lnk+\frac{c}{6}ln{\tau }_Q.$ In consistency with the exact $c=\frac{1}{2}$, the best fits yield $c=0.524\left(3\right),0.5030\left(4\right),0.478\left(2\right)$ at $t/\widehat{t}=-1,0,1$, respectively. The plots in all three panels collapse asymptotically with increasing ${\tau }_Q$, demonstrating the space-time scaling (75) for large enough ${\tau }_Q$. The collapsed plots for large ${\tau }_Q$ are the scaling function $F_S(t/\widehat{t},L/\widehat{\xi })$ in Eq. (75).

Figure 11

Entanglement gap in Eq. (77). The left, middle, and right panels show the gap before ($t/\widehat{t}=-1$), at ($t/\widehat{t}=0$), and after ($t/\widehat{t}=+1$) the critical point, respectively. Different colors of the plots correspond to different quench times ${\tau }_Q$. All plots are rescaled: they are presented as a function of the rescaled block size $L/\widehat{\xi }=L/\sqrt{{\tau }_Q}$ and the gap is multiplied by ${\widehat{\xi }}^{\beta /\nu }={\left(\sqrt{{\tau }_Q}\right)}^{1/8}$. The plots in all three panels collapse asymptotically with increasing ${\tau }_Q$, demonstrating the space-time scaling (79) for large enough ${\tau }_Q$. The collapsed plots for large ${\tau }_Q$ are the scaling function $F_{\mathrm{\Delta }\lambda }(t/\widehat{t},L/\widehat{\xi })$ in Eq. (79).